Analytic description of nanowires II: morphing of regular cross sections for zincblende- and diamond-structures to match arbitrary shapes

Setting out from König & Smith (2021), we present an analytic morphing of zincblende- and diamond-structure nanowire (NWire) cross sections to arbitrary convex shapes as encountered in experiment. We predict the NWire atoms, bonds between these atoms and NWire interface bonds, plus characteristic lengths and area of the NWire cross sections. Cross section areas, ratios of internal bonds per NWire atom enable an interpretation of any spectroscopic NWire data. Our algorithms include a radial dependence of the NWire unit-cell parameter and can be applied to multi-core-shell NWires where NWire layers can be morphed independently from each other.


Introduction
In recent publications, we derived (Kö nig & Smith, 2019) and improved  the analytical description of six regular zb-NWire cross sections relevant to experiment (Weber & Mikolajick, 2017); see Fig. 1. To this end, we described the number of atoms, the number of bonds between such atoms and the number of interface bonds for an NWire slab with a thickness of the periodic unit cell (UC) along its growth axis with its interface length, height, width and NWire cross section area. An analytical structural description of the NWire cross section down to the individual bond and atom is a powerful tool for interpreting or predicting  any experimental data as a function of NWire cross section size, shape and orientation of its growth axis and interfaces. Here, we aim to extend this analytic description for zb-and diamond-structure NWires to arbitrarily convex cross sections featuring linear interfaces, thereby allowing one to fit the analytics of such cross sections to any irregular convex shape encountered in experiment.
Section 2 provides the necessary background information on the nomenclature on how to interpret the cross section images per NWire type, and a brief assignment of primary and secondary parameters to structure-driven phenomena. Section 3 contains the number series of all six different NWire cross sections, as shown in Fig. 1 for uniaxial morphing (C 2 symmetry). In Section 4, we introduce triaxial morphing to all four hexagonal cross sections (C 3 symmetry) with three independent run indices, allowing for a vast range of cross section shapes. Combining the morphing algorithm in both sections, virtually any crystalline zb-NWire with convex cross section geometry can be described. In Section 5, we show examples of applying the number series and derived secondary parameters to experimental data from the literature for each, irregular Si and core-shell III-V NWires. Appendices A, B and C derive characteristic lengths and areas for cross sections with [110], ½11 " 2 2 and [111] growth vectors, respectively. As this work builds upon our previous publications, we refer the reader to Kö nig (2016) and  for the background information on chosen cross sections, interface energetics, bond densities and further details regarding the basics of associated analytic number theory.
2. General remarks on analytic number series, structural boundary conditions and nomenclature Table 1 lists the primary and secondary parameters calculated by number series.
All parameters are calculated over an NWire slab presenting the thickness of the UC a uc in the growth direction as per  to achieve periodicity (Table 2). In addition, Table 2 lists the amount of atoms and bonds per column (i.e. per atom or bond visible) as a function of NWire axis orientation for a top view onto the cross section, thereby allowing atoms and bonds to be counted. Respective images are provided for all NWire cross section types presented here.  Table 2 Slab thickness d slab of NWire cross sections as a function of the growth axis orientation given in UC lengths a uc per growth orientation to achieve periodicity. Numbers of atoms and of bonds per column as described per feature seen in cross section top view are given to enable the counting of atoms and NWire-internal bonds.  The periodicity in the growth direction and the assumption that the length of the NWire l Wire exceeds its diameter d Wire allows for a highly accurate description of parameters, though, as per mathematical definition, they are correct only for l Wire =d Wire ! 1.
On a par with , the indexing of NWire cross section type is given as a superscript with its shape and growth direction; see Table 3.
With N bnd =N Wire , we obtain a gauge for the response to internal stress, e.g. by dopant species. The ability of embedding materials or ligands to exert stress (Schuppler et al., 1994;Boyd & Wilson, 1987) onto NWires or vice versa can be described with N IF =N bnd . The impact of a highly polar surface termination on the zb-NWire electronic structure observed as interface-related electronic phenomena (Zahn et al., 1992;Campbell et al., 1996;He et al., 2009;Kö nig et al., 2014Kö nig et al., , 2018Kö nig et al., , 2019 is assessed by the ratio N IF =N Wire . The ratio N abc IF =N def IF can be useful for detecting facet-specific interface defects. For Si, interface-specific dangling bond (DB) defects exist, namely, the P b0 centre at {001} interfaces and the P b1 centre at {111} interfaces (Helms & Poindexter, 1994;Keunen et al., 2011). These DB defects occur in a ratio which reflects N 001 IF =N 111 IF and can be detected by electron paramagnetic resonance (EPR) (Stesmans et al., 2008). For Si-NWires, the ratio N abc IF =N def IF is therefore a valuable tool for identifying cross sections of the NWires treated in Sections 3.2, 4.1 and 4.2. We illustrate the results on tetrahedral C, Si and Ge NWires (all diamond structure). NWire atoms without interface bonds are shown in grey. Atoms with interface bonds are colour-coded: species with one/two/three interface bonds are red/blue/green; see Fig. 1 for an example. The analytical number series introduced below also hold for zb-NWires due to straightforward symmetry arguments . Material properties resulting from differences in the base cell -A-B for zb-structures versus A-A for diamond structures -are not considered here. This constraint has no impact on the applicability of the analytics of our work, unless the atomic sequence mentioned above is of primary interest when comparing two solids.
The nominal number series describing the high-symmetry NWire cross sections follow a run index i which defines the nominal size of the cross section. Morphing of cross sections is introduced by a second class of run indices j 1 ; j 2 (j; k 1 ; k 2 or k 1 ; k 2 ; k 3 ) for C 2 symmetry uniaxial (C 3 symmetry triaxial) morphing, defining the shape -or more precisely, its deviation from the respective high-symmetry cross section. For quandrangle cross sections treated in Sections 3.1 and 3.2, one index j is sufficient to describe the symmetry deviations elaborated here, as is straightforward to see by turning cross sections by 90 . For the four remaining more complex hexagonal cross sections, we introduce two running indices j 1 ; j 2 to allow for independent morphing from the top and bottom interfaces. Generally, we have j ¼ j 1 ¼ j 2 ¼ 0 for the nominal shape of the cross section. The morphing indices then span the range of j ¼ i; i À 1; . . . ; 0; À1; . . . ; Ài; . . . ! À1 for quadrangle cross sections, and -with one exception (see Section 3.5) -of j ¼ i À 1; . . . ; 1; 0; . . . À ði À 1Þ for hexagonal cross sections, the positive limit of the latter occurring due to their interface planes intersecting at finite distance for i < 1 (as opposed to parallel interfaces for quadrangle cross sections). An example of cross section morphing is shown in Fig. 2. These limits to lateral run indices for hexagonal cross sections are also valid in triaxial morphing with lateral run indices j; k 1 ; k 2 or k 1 ; k 2 ; k 3 (Sections 4 and 5.1), again with one exception (Sections 4.3 and 5.2).
For the hexagonal cross sections, we originally developed an even and an odd series to account for minor deviations from the high-symmetry cross sections in experiment . The differences between parameters from odd versus even series are outrun by far with the modifications due to morphing. As a consequence, we introduce morphing here only to the even series of all hexagonal cross sections. While adequate number series modifications can also be derived for the odd series of all hexagonal cross sections, their descriptions of NWire cross sections are covered by morphing the even series onto experimental data. An example is the cross sections of Si NWires with a [110] growth axis and {001} plus {111} interfaces; see Fig. 1(c) herein and the experimental data published by Yi et al. (2011). This cross section was treated in Kö nig & Smith (2019) with even and odd cross section calculus. With the morphing introduced in Section 3.3 and, in particular, in Section 4.1, we can simply use the even series and morph it exactly onto the experimental image.

Morphing cross sections along one symmetry axis
Terms describing the high-symmetry cross section (i.e. j ¼ j 1 ¼ j 2 ¼ 0) are identical with the respective Equations in . Such terms are printed here in grey to distinguish them from terms due to morphing which are printed in black.

Nanowires growing along the [001] direction with square cross section and four {001} interfaces
As mentioned briefly in the Introduction, run indices for this cross section are limited to i ! 1 and À1 < j < i.
The square shape of the cross section results in w½i; j h½i; j d 001Àu t 001ÀIF ½i; j. As mentioned briefly in the Introduction, run indices for this cross section are limited to i ! 1 and À1 < j < i.
Since we morph the cross section along the {110} interfaces, N 110À u t 001ÀIF ½i ¼ 8ði þ 3Þ does not change with j. However, is a function of j by which the ratio of interface bonds between facets becomes The centre expression shows both number series in their explicit form, while the expression on the right presents the simplified result of their ratio. As was the case for N 110À u t 001ÀIF ½i, the length of the {001} interface remains unchanged;  Cross section of the zb-/diamond-structure NWires growing along the [001] axis with a square cross section and four {001} interfaces for run index i = 3 and (a) expansive morphing with index j = À3, and (b) reductive morphing with index j = 2. The latter morphing is not considered useful since a 90 rotation yields to expansive morphing at a lower run index i, thereby restricting j to negative integers (here: i = 1 and j = À2). Yellow atoms show the outer limit of the nominal cross section (i = 3 and j = 0) in part (a) and white atoms in part (b) present 'ghost atoms' to fill up the nominal cross section. Interior atoms are grey, atoms with two interface bonds are blue and atoms with three interface bonds are green.
The width of the rectangular cross section follows in a straightforward manner from w½i d 110À u t 001ÀIF ½i. Morphing has an impact on the length of the {110} interfaces, resulting in The height of the rectangular cross section follows in a straightforward manner from h½i; j d 110À u t 110ÀIF ½i; j. The total cross section area is given by The cross section of this NWire type is shown in Fig. 4.

Nanowires growing along the [110] direction with a hexagonal cross section and four {111} plus two {001} interfaces
The remaining four NWire types to be investigated all have a hexagonal cross section which has a more complex geometry. As mentioned briefly in the Introduction, run indices for these cross sections are limited to i ! 1 and Àði À 1Þ j i À 1, except for the cross section with an exclusive {110} interface and a [111] growth axis; see Section 3.5.
For the number of atoms forming the NWire cross section, we get The number of bonds between these atoms is described by The number of interface bonds over all facets is given by Since the number of {111} facet bonds being added or removed equals the number of {001} facet bonds being removed or added per change in j 1 or j 2 , such contributions cancel each other out; see Equation 17 below. For a graphical verification, we refer the reader to Fig. 5, or to Fig. 4 in Kö nig & Smith (2021). The ratio of interface bonds per facet orientation is given by The lengths of the {001} and {111} facets depend only on one lateral run index j ; 2 1; 2, which affects the respective facet. For the {001} facet, a scaled offset of 1 exists for the two irregular triangular areas in the apexes: For the {111} facet, a scaled offset of À1/4 exists due to the two irregular triangular areas in the apexes: Due to morphing along the vertical symmetry axis of the cross sections, w 110À ½i 6 ¼ f ðjÞ and thus stays unchanged. The scaled offset due to the two isosceles triangles at the side apexes of Cross section of the zb-/diamond-structure NWires growing along the [110] axis with a rectangular cross section and two {001} interfaces at the top and bottom, and two {110} interfaces at the sides, for run index i = 3, showing (a) expansive morphing with morphing index j = À6 and (b) reductive morphing with j = 2. Yellow atoms show the outer limit of the nominal cross section (i = 3 and j = 0) in part (a), white atoms in part (b) present 'ghost atoms' to fill up the nominal cross section. Interior atoms are grey, atoms with one interface bond are red, atoms with two interface bonds are blue and atoms with three interface bonds are green. Due to two different interface orientations, j > 0 is useful to calculate expansive morphing in the horizontal direction (i.e. parallel to the {001} interfaces) by picking an appropriate i to match d 110À u t 001ÀIF ½i and then reducing the vertical extension by j > 0. the cross section is ffiffi ffi 2 p =8 which is added to the nominal increment of ffiffi ffi 2 p ði À 1=2Þ: Obviously, the height of the cross section does change with j 1 ; j 2 in steps of Ç a uc =2 per j ! j AE 1, resulting in The total cross section area is described by The prefactor a 2 uc = ffiffi ffi 8 p presents the area of one X 6 ring, seen along the h110i lattice vector, which is straightforward to derive from four such rings filling the zb-UC cross section when cut along the {110} plane, covering an area of ffiffi ffi 2 p a 2 uc ; see Appendix A. The offset areas concern the isosceles triangles at the {001} facets with an area of a 2 uc ffiffi ffi 2 p =16, and the irregular triangles at the {111} facets with a 2 uc ffiffi ffi 2 p =64, both of which can be found when considering an X 6 ring seen along the h110i lattice vector, using geometrical arguments. The total offset area 4/16+2/32 presents the four scalene triangles at the two lower and upper apexes of the cross section, plus the two isosceles triangles occurring at the left-and rightmost apexes of the cross section 1 , see Figs. 5 and 16, Appendix A and Fig. 4 in Kö nig & Smith (2021).

Nanowires growing along the [11 "
2 2] direction with hexagonal cross section and four { " 1 131} plus two {111} interfaces For the number of atoms forming the NWire cross section, we get The number of bonds between these atoms is described by The number of interface bonds over all facets is given by The assignment of interface bonds to the {111} and {1 " 3 31} facets is shown in Fig. 10  Cross section of the zb-/diamond-structure NWire growing along the [110] axis with a hexagonal cross section and four {111} plus two {001} interfaces for run index i = 6, shown with axial-symmetric morphing which is expansive on top with index j 1 = À3, and reductive at the bottom with index j 2 = 4. Translucent lines show the outer limit of the cross section and respective facet lengths. Observe the irregular triangular areas at the apexes of the cross section which are a constant offset to the total cross section area, its width and facet lengths. For atom colours, refer to Fig. 4. For a detailed geometrical derivation of characteristic lengths and areas, refer to Appendix A.
Equation 26 shows the explicit number series in the top line, while the bottom line is the compacted version for the ratio of interface bonds. The following facet lengths depend only on one lateral run index j ; 2 1; 2, which is assigned to the facet of interest: and For the f1 " 3 31g facets, the smallest unit is the diagonal of the congruent rectangular areas constituting the cross section area. These rectangles have a horizontal scaled length of 1= ffiffi ffi 8 p (see Equation 29) and a vertical scaled length of 1= ffiffi ffi 3 for the scaled diagonal of the rectangle. Due to morphing along the vertical symmetry axis of the cross sections, w 11 " 2 2À ½i 6 ¼ f ðjÞ and thus stays unchanged: The height of the cross section obviously changes with morphing, following The length a uc = ffiffi ffi 3 p presents a third of the diagonal connecting two opposite corners in h111i direction through the zb-UC, whereby the (111) vector is orthonormal to the f11 " 2 2g plane; ð111Þ ? f11 " 2 2g. This length is equivalent to the longer side of the rectangle which presents the unit area of NWires growing along the h11 " 2 2i vector class, accounting for the increment in h 11 " 2 2À in Equation 30; see Fig. 2. The total cross section area is described by The scaled coefficient of 1= ffiffiffiffiffi 24 p describes the rectangular unit area of the cross section as discussed above, following from 1= ffiffi ffi 8 p . Facets cut the outermost rectangles along their diagonal, rendering their triangular area 1=ð2 ffiffiffiffiffi 24 p Þ.

For an illustration of the morphed hexagonal cross section with {111} top and bottom interfaces plus {11 "
3 3} side interfaces, refer to Figs. 2 and 10. For a detailed geometrical derivation of characteristic lengths and areas, see Appendix B.

Nanowires growing along the [111] direction with a hexagonal cross section and six {110} interfaces
The smoother geometry of this cross section allows us to use lateral run indices of Ài j i for morphing.
For the number of atoms forming the NWire cross section, we get The number of bonds between these atoms is described by The number of interface bonds over all facets is given by The facet lengths depending on the respective j ; 2 1; 2, are for the top and bottom facets, and for the side facets. The scaled coefficient 1/ ffiffi ffi 6 p refers to the side length of the equilateral triangles which form the unit area unit on a {111} plane defining the cross section. This coefficient follows from a {111} plane cut through the zb-UC along its corner points, resulting in an equilateral triangle of scaled side length ffiffi ffi 2 p , containing an area equivalent to 12 small equilateral triangles (6 equilateral + 6 isosceles with same area = 12) with a scaled side length of 1/ ffiffi ffi 6 p . The width of the cross section is not a function of j and thus stays unchanged: The height of the cross section depends on j 1 and j 2 as it is parallel to the symmetry axis along which axial-symmetric morphing occurs: The total cross section area is described by The scaled coefficient of ffiffi ffi 3 p /24 describes the area per equilateral triangle as the unit area of the cross section and follows directly from our discussion of facet lengths above. Fig. 6 shows the cross section of this NWire type.

Nanowires growing along the [111] direction with a hexagonal cross section and six {11 " 2 2} interfaces
This cross section returns to the nominal limitation of lateral (morphing) run indices, i.e. Àði À 1Þ j i À 1 with 2 1; 2. For the number of atoms forming the NWire cross section, we get The number of bonds between these atoms is described by The number of interface bonds over all facets is given by The facet lengths of top and bottom interfaces depend on the respective j ; 2 1; 2: The facet length of side interfaces is Since the width of this hexagonal cross section is not a function of j , it remains unchanged: For the height of the cross section, we get The cross section plane has the same orientation as in Section 3.5, with the facet orientation rotated by 60 ({110} ! f11 " 2 2g). This rotation swaps the scaled coefficients of facet lengths and cross section width on one side and cross section height on the other side when compared to Section 3.5 (see discussion there).
The total cross section area has the same scaled coefficient of ffiffi ffi 3 p =24 as in Equation 39 due to the same orientation of the NWire cross section (same growth vector) and thus the same small equilateral triangles as the unit area: Axial-symmetric morphing of the zb-/diamond-structure NWires growing along the [111] axis with a hexagonal cross section and six {110} interfaces. The nominal cross section for i = 8 is shown by white 'ghost atoms' at the top interface and by the yellow atoms in the lower half of the cross section. Reductive morphing with j 1 = 2 was applied to the top interface, while morphing to maximum expansion with j 2 = Ài = À8 was applied to the bottom interface. For atom colours, refer to Fig. 4. For a detailed geometrical derivation of characteristic lengths and areas, refer to Appendix C.
The outermost area elements at the facets form isosceles triangles (Fig. 7), which have the same area as their equilateral counterparts mentioned above; see also the related discussion in Section 3.5 and the geometrical derivation explained in Appendix C. The cross section of this NWire type is shown in Fig. 7.

Morphing cross sections along three symmetry axes
Such morphings naturally lend themselves to cross sections with hexagonal symmetry. We therefore do not consider square cross sections with h001i normal vectors on growth plane and facets, as well as rectangular cross sections with h110i growth vector and {001} and {110} facets. Depending on the symmetry of the hexagonal cross section, we have to introduce different lateral number series per facet orientation, with run indices j as used in Section 3, and run indices k 1 ; k 2 for the other two morphing directions, with a different facet orientation {abc} for both run indices k 1 and k 2 . This is the case in Sections 4.1 and 4.2.
As in Section 3, lateral run indices -j; k 1 ; k 2 ; k 3 -are positive for reductive morphing (cutting into the nominal cross section) and negative for expansive morphing (extending the nominal cross section), with the nominal cross section presented if all lateral run indices are zero. Under the condition that all lateral run indices are equal, i.e. j ¼ k 1 ¼ k 2 , or k 1 ¼ k 2 ¼ k 3 , all cross sections will assume a triangular or quasi-triangular shape on maximum expansive morphing, as well as on maximum reductive morphing; see Fig. 8(a).
Other, more irregular, shapes can be described in an arbitrary fashion, under the constraint that all facets or singular points where facets meet do not penetrate the nominal hexagonal cross section. As a result, all facets not being directly morphed via a lateral run index have a minimum length or at least a point where adjacent (directly morphed) facets meet. These minimum lengths or singular points are all located on the respective borders (facets) of the nominal cross section considered. By preventing the penetration of such minimum facet lengths or singular points into the nominal cross section, we prevent the lateral number series from overlapping with each other, erasing the facet between the two associated morphing sections in the process. Thereby, we obtain a minimum facet length or a common point between two neighbouring morphed facets for maximum reductive morphing. A minimum facet length refers to cross sections morphed in Sections 4.1, 4.2 and 4.4, and a common point between two neighbouring facets refers to morphing in Section 4.3. While such overlap can be dealt with in number theory and crystallography, we point out that -besides its Axial-symmetric morphing of the zb-/diamond-structure NWires growing along the [111] axis with a hexagonal cross section and six {112} interfaces. The nominal cross section for i = 5 is shown by white 'ghost atoms' at the top interface and by the yellow atoms in the lower half of the cross section. Maximum reductive morphing with j 1 = 4 was applied to the top interface, while morphing to maximum expansion with j 2 = À(i À1) = À4 was applied to the bottom interface. For atom colours, refer to Fig. 4. For a detailed geometrical derivation of characteristic lengths and areas, refer to Appendix C.  complexity -such a description of NWire cross sections is not beneficial since the free choice of the nominal run index i per cross section and subsequent morphing within these constraints covers virtually any convex NWire shape encountered in experiment. Apart from Section 4.3, where we introduce slightly different limits on run indices to prevent an overlap, such limitations are as follows. All lateral run indices have a minimum value of j ¼ k 1 ¼ k 2 ¼ Àði À 1Þ, resulting in maximum expansive morphing; see Figs. 9(a), 10(a) and 12(a). Run index doublets are limited to j + k 1 = j + k 2 = k 1 + k 2 = i À 1 in Sections 4.1 and 4.2, and to k 1 + k 2 = k 2 + k 3 = k 3 + k 1 = i À 1 in Section 4.4. The cross section treated in Section 4.3 has a limit on run index doublets of k 1 + k 2 = k 2 + k 3 = k 3 + k 1 = i, because its high symmetry and atom interconnectivity at the corner points allows for facets not being directly morphed to be reduced to singular points on the boundary of the regular cross section (versus minimum facet length for all other three cases). The basic principle of 3-axes morphing and related implications for overlap in size and form of cross sections is depicted in Fig. 8.
All number series reflect the variables we presented in Section 3, with additional series for facet lengths, NWire widths and heights, which depend on lateral (morphing) run indices. These are required in particular for finding the right indices to fit experimental values, such as facet lengths, heights or widths of NWire cross sections; see Section 5. As mentioned before in Section 3, the contribution of the nominal cross section to the respective number series is printed in grey to facilitate the decomposition into contributions per run index. For the same reason, most number series will be shown uncompacted, followed by their shortest form.

3-Axes morphing of nanowires growing along the [110] direction and four {111} plus two {001} interfaces
For the number of atoms in the NWire cross section, we obtain The number of bonds between these NWire atoms are described by The total number of interface bonds of the NWire cross section amounts to As was the case with axial-symmetric morphing (see whereby the top row in Equation 51 show the explicit number series per interface orientation and the lower row presents their ratio. The length of the top f001g facet is given by whereby the analogy of Àk 1 À k 2 to À2j in clearly visible; see Equation 18. The length of the bottom {001} facet is given by being equivalent to Equation 18. The length of the two upper {111} facets depends only on the respective k ; 2 1; 2: In Equation 54, we add or remove one X 6 ring per change in k , as is the case for j in Equation 19, underlining the high symmetry of the NWire cross section. The length of the two lower {111} facets depends on the respective k and j, the latter limiting such facets from below: These facet lengths are shown in Fig. 9(b). Due to d 110À 001ÀIF;bot ½i; j w 110À ½i; k 1 ; k 2 , the width of the cross section depends on k 1 ; k 2 only, which is a direct consequence of the morphing limits discussed at the beginning of Section 4: For the same reason, the height of the cross section depends only on j: The total area of the cross section naturally depends on all running indices: The underbrace in line 2 of Equation 58 indicates the offset area which is composed of four scalene and two isosceles triangles at the corner points of the cross section; see Appendix A and Fig. 16 for their derivation 2 . The underbraces in line 3 of Equation 58 denote the contribution to maximum extensive morphing per class of lateral run indices j and k 1 ; k 2 , from which the respective area is subtracted when j; k 1 ; k 2 > À ði À 1Þ. Fig. 9 shows crystallographical details of this cross section and a couple of examples of triaxial morphing.

131} plus two {111} interfaces
For the number of atoms in the NWire cross section, we obtain The number of bonds between these NWire atoms are described by The total number of interface bonds of the NWire cross section amounts to The assignment of interface bonds to the {111} and {1 " 3 31} facets is shown in Fig. 10(b). With this assignment of interface atoms to {111} and {1 " 3 31} facets, we obtain for the respective explicit number series in the top row and for the more compact form describing the ratio of interface bonds only. There are four different facet lengths which have to be used with their respective run indices j and k , 2 1; 2, as required for the facet of interest: For an illustration of facet lengths, we refer to Fig. 10. As discussed in Section 4.1 around Equation 56, the width of the cross section depends on k 1 ; k 2 only: For the same reason, the height of the cross section depends only on j: The total cross section area is presented by þiði À 1Þ |fflfflfflffl ffl{zfflfflfflffl ffl} max: ext: morphing with j þ 2ði À 1Þði þ 1Þ |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl ffl} max: ext: morphing with k 1 ;k 2   (2021)] is shown by purple atoms and the corresponding maximum expansive morphing by cyan atoms. (b) Cross section with i = 5, j = 0, k 1 = À2 and k 2 = À4, with magenta lines assigning interface bonds to respective facets, and green arrows and labels showing interface lengths. For an example of reductive 3-axes morphing, refer to Fig. 2(c). For a detailed geometrical derivation of characteristic lengths and areas, refer to Appendix B.
As for Equation 58, we have assigned the contribution to maximum extensive morphing per class of lateral run indices j and k 1 ; k 2 from which the respective area is subtracted when j; k 1 ; k 2 > À ði À 1Þ, before converting Equation 69 to its shortest form. For an illustration of irregular 3-axes morphing of this cross section, refer to Fig. 10.

3-Axes morphing of nanowires growing along the [111] direction with a hexagonal cross section and six {110} interfaces
This cross section has a higher symmetry, as opposed to the two previous cases in Sections 4.1 and 4.2. All three morphing areas are identical and subject to their respective run index, which becomes apparent if we look at their interface orientations, which are identical to each other. As a result, we introduce just one class of run indices k 1 ; k 2 ; k 3 . We also point out that the morphing areas are identical to those depending on j in Section 3.5. The difference occurs by the morphing of opposite areas (referring to a C 2 symmetry), while here we morph three equal areas -subject to identical run indices k ; ¼ 1; 2; 3 -when rotated by 120 (C 3 symmetry). As there is no overlap in morphing regions for one k ¼ i under the constraint that the other two k indices are 0 [see Fig. 11(c)], and the ultimate corner point of the extensive morphing occurs for k ¼ Ài [see Fig. 11(a)], we can extend the k all the way to AEi. Still, the indices k 1 ; k 2 and k 3 are restricted over ½Ài; þi along k þ k i, where ½; are cyclic permutations of the run indices, viz. ½1; 2; ½2; 3; ½3; 1. Thereby, we avoid the morphing of the three triangular areas running into each other. As examples, if i = 9 and k 1 = 9, we have Ài k 2 + k 3 0; if i = 9 and k 2 = 7, we have Ài k 3 + k 1 2, etc.
For the number of atoms in the NWire cross section, we obtain The final form of Equation 70 summarizes the terms which depend on k into a sum for brevity; we will use this short form in all subsequent equations where applicable. The number of bonds between these NWire atoms is described by The total number of interface bonds of the NWire cross section amounts to There are two types of interface lengths. One represents the facets normal to the morphing vector given by atomic planes being added or subtracted, and depends only on the respective k :  Another interface length exists for facets which are modified in their length by the two adjacent morphing regions: Due to the C 3 symmetry of the hexagonal cross section, its height can be calculated along all three morphing vectors: The calculation of the cross section width does not appear to be useful. It would require a discrimination to depart from the nominal width when k ; k < Ài=2, adding an increment of Àa uc = ffiffi ffi 6 p 1 2 ½k þ k , which is somewhat cumbersome in handling experimental data. We therefore rely onto the height of the cross section as per Equation 75, which should be sufficient to assign run indices to an experimental image.
The total cross section area is presented by The maximum external morphing per k assigned in the top row of Equation 76 is straightforward to see in Fig. 11(a), where the three equilateral triangles cover half of the nominal cross section consisting of six of such triangles. Morphing examples are shown in Figs. 11(b) and 11(c).

3-Axes morphing of nanowires growing along the [111] direction with a hexagonal cross section and six {11 " 2 2} interfaces
This cross section reverts back to the restrictions on the run indices Àði À 1Þ k 1 ; k 2 ; k 3 i À 1 we had for cross sections in Sections 4.1 and 4.2, together with the restriction k þ k i À 1, where ½; are cyclic permutations of the run indices; see beginning of Section 4. The symmetry considerations given in Section 4.3 also apply to this cross section which has exclusive {11 " 2 2} facet orientations. For the number of atoms in the NWire cross section, we obtain The number of bonds between these NWire atoms are described by By analogy to Equations 73 and 74, we have two types of facet lengths with their respective lateral run indices, namely, the facets normal to the morphing vector given by atomic planes being added or subtracted and for facets which are modified in their length by the two adjacent morphing regions: As was the case in Section 4.3, the C 3 symmetry of the hexagonal cross section allows for its height to be calculated along all three morphing vectors: The total cross section area is presented by þ 3ði 2 À 1Þ þ 3ði 2 À 1Þ þ 3ði 2 À 1Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} max: ext: morphing with k 1 ;k 2 ;k 3 The maximum external morphing per k assigned in the top row of Equation 83 is straightforward to see in Fig. 12(a), where the three axial-symmetric trapezoids fold back onto the nominal cross section, which consists of six such trapezoids, whereby the two central atoms shown in black in Fig. 12

Si NWires
Si NWires have been shown to grow as monolithic crystals along the [111] axis with atomically flat f11 " 2 2g interfaces when aluminium (Al) is used as a seed catalyst (Moutanabbir et al., 2011). Such NWire cross sections are shown in Fig. 13. We picked two examples from this reference to show the usage and results derived by the morphing algorithms from experimental input. Table 4 shows all parameters and results of the cross sections shown in Figs. 13(b) and 13(c), respectively.
Due to several run indices present, the fitting onto the exact cross section shape requires an iterative process which is well suited to a computer code. Such a code could be added to existing visual software for gauging NWire cross sections -a task we illustrate here in a stepwise fashion as a principal guide. As unit-cell parameter for Si, we use a uc = 0.54309 nm (Bö er, 1990).
There are two ways to start an iteration for obtaining the run indices. The first is to start with one d 111À j110 IF; non5 ði; k ; k Þ (Equation 74) and its two adjacent d 111À j110 IF; 5 ði; k Þ (Equation 73), rearranging for the three run indices such as i; k 1 ; k 2 . This approach may be more appealing to the experimentalist, and is illustrated on a core-shell NWire in Section 5.2. The more direct starting point for the iteration is given by using h 111À j110 ði; k Þ (Equation 75) and the d 111À j110 IF; 5 ði; k Þ, which is at one end of h 111À j110 ði; k Þ, rearranging only for the two run indices involved.
With the measured height and interface lengths h and a, as listed for Fig. 13(b) in Table 4 yielding k 3 ¼ À24 . . . À 22 and i = 127. The tolerance range for k 1 ; k 2 and k 3 originates from the tolerance in length measurement which is in the range AE1 nm; see Fig. 13 and Table 4. Such tolerance ranges translate into ranges for k and i which serve to minimize the difference to the respective calculated length jd 111À j110 IF ði; k ; k Þ À Xj; see Equation 85 below and Equation 86 in Section 5.2. With i known, we can proceed with Equation 84(II) to get a start range for the other two k , yielding k 2 (i = 127, c) = À19 . . . À 17 and k 1 (i = 127, e) = À24 . . . À 22. Next, we rearrange Equation 74 for its run indices, viz.
Equation 85 provides us with a sum of k þ k ¼ P k, which we can match under consideration of the range of k and k . For d 111À j110 IF; non5 ði; k 1 ; k 2 Þ ¼ f ðX ¼ dÞ, we get P k = À43 . . . À41 = k 1 + k 2 . This range allows for k 1 ¼ À23 . . . À 26 and k 2 = À17 . . . À19. Moving on to d 111À j110 IF; non5 ði; k 2 ; k 3 Þ = f(X = b), we obtain P k = À43 . . . À41 = k 2 + k 3 . This range allows for , the k indices get narrowed down towards one integer value. We can narrow down the range for the k further by reconsidering Equation 84(I) with i = 127 and k 3 ¼ À22 . . . À 20 to match the experimental value h [ Fig. 13(b) and Table 4]. This is best achieved using h 111À j110 ði ¼ 127; k 3 = À22) = 91.8 nm, leaving just 0.4 nm to match the measured value of h. With k 3 = À22, we can iterate again, obtaining k 2 = À19 and k 1 = À23 AE 1 = À23, whereby we chose the centre of the k 1 range to arrive at a minimum deviation from the measured length parameters of the NWire cross section. An iterative computer code would modify k 1 ; k 2 and k 3 around their initial values such that the sum of the absolute deviation values of all the NWire length parameters from their measured counterparts jd 111À j110 IF ði; k ; k Þ À Xj is minimized as the criterion to arrive at the best structural fit of the number series; see     Table 4 for the parameters of the respective NWire cross section. Equation 86 in Section 5.2. The indices [i, k 1 , k 2 , k 3 ] = [127, À23, À19, À22] can then be used in Equations 77 to 83 to calculate the structural results. These are shown in Table 4, together with the results for the cross section in Fig. 13(c).
The flexibility of the above algorithms in describing the NWire cross section could also be very useful for NWire shapes changing by post-growth extrinsic means. As an example, an atomic rearrangement at Si NWires due to high current densities inducing local heating (Bahrami et al., 2021) can be tracked and linked to the surface energies of respective facets. Tracking such changes over time with our crystallographic description and the energy intake by local heating should allow the atomic surface diffusion process to be described in much detail. Such findings can be key to NWire design on demand.
5.2. Core-shell III-V zb-NWires with different unit-cell parameters a uc III-V NWires are often found to grow along the [111] axis which requires the least energy and have hexagonal cross sections (Joyce et al., 2011;Treu et al., 2015). Here we will focus on core-shell GaAs-In 0.2 Ga 0.8 As zb-NWires grown by solid-state molecular beam epitaxy (MBE) along the [111] axis with {110} interfaces, using visual and crystallographic data from Balaghi et al. (2019). Fig. 14 shows the NWire cross section with crystallographic details and the assignment of variables to respective interface lengths and one UC height of the NWire cross section.
Below, we will show how we can derive structural results using Equations 70 to 76 in Section 4.3 to match all interface lengths to their measured value (Table 5), taking the different unit-cell parameters for core and shell of NWire, a core uc and a shell uc , into account. The resulting run indices allow us to obtain the main variables N 111À j110 Wire , N 111À j110 bnd , N 111À j110 IF and A 111À j110 for the core and total NWire cross sections, hereafter denoted as N Wire , N bnd , N IF and A, respectively, to keep the prsentation of variables as simple as possible. The same simplification goes for all d 111À j110 IF and h 111À j110 IF , hereafter denoted as d IF and h, respectively. From these data, we can derive the main variables of the shell by simple differential/ additive calculations. The calculation of run indices for the core and shell of the NWire cross section require additional indexing of the run indices to avoid confusion: run indices using the unit-cell parameter of the NWire core a core uc will be i core ; k core 1 ; k core 2 ; k core 3 , and run indices using the unit-cell parameter of the NWire shell a shell uc will be i shell ; k shell 1 ; k shell 2 ; k shell 3 . We set out with the cross section of the NWire core (Fig. 14) and use the lengths of three adjacent interfaces as a starting point, thereby illustrating the second method briefly mentioned in Section 5.1 of how to find the run indices i core , k core 1 , k core 2 , k core 3 NWire cross sections. The convergence criterion (CC) we use for obtaining the run indices which describe the NWire cross section with minimum deviation is given by the sum of absolute deviations of the calculated d IF; non5 , d IF; 5 and hði; k Þ -see Equations 73 to 75 in Section 4.3 -from their respective measured values a to f plus h 1 to h 3 for the NWire core, and A to F plus H 1 to H 3 for the NWire shell; see Fig. 14(c): Please observe that below we will work with only one height per cross section to keep the explanation of the fitting procedure concise and Fig. 14(c) readable; the inclusion of all three heights into a computer code featuring Equation 86 is straightforward. As a starting point for all cross sections, we set k 1 ¼ k 2 ¼ k 3 = 0 (presenting the nominal regular shape of the NWire cross section) and run an iteration scheme with Equation 86 in compound with Equation 87, using i as a run index to arrive at an educated guess from where to start the fitting procedure.
Our search criterion features a length of an interface which depends on three run indices, d IF; non5 ½i core ; k core ; k core , plus its adjacent interface lengths d IF; 5 ½i core ; k core and d IF; 5 ½i core ; k core . We start with choosing core = 1 and core = 2 to obtain the following convergencies: d IF; non5 ½i core ; k core 1 ; k core 2 ! d, Kö nig and Smith Analytic description of nanowires II 659 Table 5 Parameters for hexagonal core-shell NWire cross section as per Fig. 14 Table 5: , thereby eliminating all k , yielding the search condition for a regular hexagonal cross section to make an educated guess at a starting value for i core . The unit-cell parameter we use is the value for GaAs, a core uc = 0.56533 nm (Bö er, 1990), which comprises the NWire core; see Fig. 14. If i is located nearly halfway between two integer values, we may run the calculations below with both adjacent i values to see which one has the lower CC as per Equation 86. Once we got i core , we can run each single Equation 87 (I) to (III) for obtaining k core 1 and k core 2 . We then move on to the next k core and k core , namely, k core 2 and k core 3 , and to the next three measured interface lengths e, f and a. The third index rotation then uses k core 3 and k core 1 with the interface lengths a, b and c. As a result, we obtain the final results for the NWire core, N core Wire ða core uc Þ, N core bnd ða core uc Þ, N core IF ða core uc Þ and A core ða core uc Þ. When iterating for the NWire shell cross section, we proceed the same way, using the adequate variables as mentioned above.
As in Section 5.1, we use the tolerance in length measurement ( Fig. 14 and Table 5) to provide narrow ranges for all k and h around their calculated value, aiming for a minimum CC while keeping i constant. For the NWire core, the result of this fitting procedure with CC ! min are the run indices i core ; k core 1 ; k core 2 and k core 3 . Replacing a to e and h 1 to h 3 in the above process with A to E and H 1 to H 3 , and using a shell uc = 0.57343 nm (Adachi, 2004) as the unit-cell parameter of the shell material featuring In 0.2 Ga 0.8 As, we get i shell ; k shell 1 ; k shell 2 and k shell 3 for the NWire shell. All these indices are shown in Table 5, together with the measured interface length and height of the respective cross section [ Fig. 14(c)].
The run indices of the NWire core cross section can be used directly to calculate all of its parameters discussed in Section 4.3. Deriving the parameters of the shell and eventually of the entire NWire system requires some additional calculations. After calculating i core ; k core 1 ; k core 2 and k core 3 with a core uc ¼ a uc ðGaAsÞ, we apply an increment to i core , viz. i core ! i core þ 1, and use the same k core 1 ; k core 2 ; k core 3 for a calculation of A core ði core þ 1; k core 1 ; k core 2 ; k core 3 Þ with a core uc . We then calculate the cross sections of the core with i core and i core þ 1 and the same k core 1 ; k core 2 ; k core 3 as obtained with a core uc . The transition i core ! i core þ 1 adds one atomic ML to the cross section, accounting for the interface region between the core and shell material as seen from the NWire core using a core uc : A intern IF ða core uc Þ ¼ A core ði core þ 1; k core 1 ; k core 2 ; k core 3 ; a core uc Þ À A core ði core ; k core 1 ; k core 2 ; k core 3 ; a core uc Þ: We then iterate again as per the above description for the NWire core, but now use a shell uc ¼ a uc ðIn 0:2 Ga 0:8 AsÞ, see the third row in Table 5, obtaining i shell ; k shell 1 ; k shell 2 and k shell 3 . With this result, we again apply the increment i shell ! i shell þ 1 and use the same k shell 1 ; k shell 2 ; k shell 3 for a calculation of Aði shell þ 1; k shell 1 ; k shell 2 ; k shell 3 Þ with a shell uc ¼ a uc ðIn 0:2 Ga 0:8 AsÞ. This transition adds one atomic ML to the core cross section as well, using the unit-cell parameter of the shell material, accounting for the interface region between the core and shell material, as seen from the NWire shell using a shell uc : With both interface areas, we can now calculate their average value as the most accurate interface area we can obtain: The reason we use two descriptions of the interface area is given by the transition of the unit-cell parameters when going from the core to the shell material. This approach can be further exploited for an ML-wise increment in cross section with a uc ð; i Þ for each increment in i , which adds further precision if a radial distribution of the unit-cell parameter is known, such as in Fig. 3(a) in Balaghi et al. (2019), and is further discussed in Section 5.2.1. The indices i ; k 1 ; k 2 ; k 3 and i þ 1; k 1 ; k 2 ; k 3 of the NWire core cross section without and with one ML as interface region for each, a core uc and a shell uc , have further use for other interim data we use to arrive at our final results.
From a practical viewpoint, the calculation of N shell IF delivers two values. For spectroscopic characterization techniques where no carrier recombination is involved, such as Raman, Fourier-transform infrared (FT-IR) or electron paramagnetic resonance, the core-shell interface bonds N core IF are considered as dipoles whereby they get counted only once with the core for the complete NWire, resulting in N shell IF ¼ N tot IF a shell uc À Á . For spectroscopic characterization techniques where carrier recombination is involved, such as photoluminescence or carrier lifetime spectroscopy, the interface bonds at the coreshell interface can acquire and trap free carriers from the core, as well as from the shell. Hence, these bonds have an impact on the core and shell, whereby we count N core IF for N shell IF in addition to including them for the core: Since the number of interface bonds does not directly depend on a core uc versus a shell uc for epitaxial NWire growth, we can drop their dependence on the unit-cell parameter in Equation 93. The area of the NWire shell requires the area of the NWire core and the core-shell interface region to be subtracted from the total NWire area, viz.
The final results of the full NWire cross section -core and shell with their respective a uc -follow from the addition of results from core and shell, namely, The area of the internal interface A intern IF (Equation 90) is another final result not included in A full , since NWire interfaces behave in a significantly different manner from the core and shell regions in terms of electronic and optical properties, such as carrier transport, recombination and interface dipoles. The calculation of the dimensionless crystallographic parameters N bnd =N Wire , N IF =N Wire and N IF =N bnd , which allow for inter-NWire comparison, are straightforward. Table 5 shows all the measured lengths of the core-shell NWire depicted in Fig. 14, together with selected interim results for the cross section of the NWire core using a shell uc listed in column 3, and all final results.

Flexibility of cross section calculations for core-shell
NWires. We have briefly mentioned above that -due to i ! i+1 adding a defined ML (usually atomic ML) to the NWire cross section -we can introduce a unit-cell parameter with a radial dependence a uc ðiÞ to the entire core-shell NWire. If such a dependence is known, e.g. for the NWire shell (Balaghi et al., 2019), the precision of the NWire description can be further increased. We note here that non-radial deviations of a uc ðiÞ, such as local inhomogeneities, cannot be accounted for due to the radial layer dependence of all number series with their main run index i.
Since our analytic treatment of zb-NWire cross sections works on the basis of smallest area segments coming along with every atom and bond considered, the arrangement of the core and shell to each other is flexible over a wide range. To illustrate the implications, the NWire core does not have to be located in the centre of the NWire shell, nor does any restriction exist for the core and shell NWire cross sections to be morphed independently from each other. This finding can be verified in our above example ( Fig. 14 and Table 5). It becomes apparent from Fig. 14 that the NWire core is not aligned with the NWire shell to share the same symmetry centre. We can go further down this path and adapt the outer NWire shape to a different interface orientation, as would be the case for a core-shell NWire growing along the [111] axis, with internal {110} and external f11 " 2 2g interfaces (Fig. 15). Such an NWire cross section requires a partition into three sections, of which two are treated in accord with the core and shell sections in the above numerical example (Fig. 14). The third section describes the outermost shape of the NWire shell, where the change in interface orientation occurs. When assembling the respective variable N Wire , N bnd , N IF and A for the entire NWire, we calculate the full shell (index 'tot' in above example) of the entire NWire with outer f11 " 2 2g inter- Cross section of the core-shell NWire growing along a [111] axis. The cross section of the NWire core has {110} interfaces, is irregular (i core = 4, k 1 core = À2, k 2 core = 0 and k 3 core = À1) and located off-centre with respect to the symmetry centre of the NWire shell. The NWire shell has {112} interfaces and is regular (i shell = 12, k 1 shell = k 2 shell = k 3 shell = 0). Arbitrary cross section shapes as per individual morphing can be combined if the core and shell share the same symmetry (crystal orientation) along the growth axis. The circumference of the NWire core is highlighted by a magenta line and the corresponding regular NWire core is shown by a cyan line. faces and then simply substract the core cross section with a shell uc , yielding the above variables for the shell with different faceting at the inner and outer interfaces. Such core-shell NWire descriptions can be chained to describe multi-coreshell cross sections by repeating the calculations shown in this section for every core-shell pair.

Conclusions
Building on our previous work (Kö nig & Smith, 2021), we introduced extensions into analytical number series for zband diamond-structure NWires for adapting their cross sections to arbitrary shape (morphing), covering the following NWire cross sections: square, h001i growth axis and interfaces; rectangular, h110i growth axis and {110} plus {001} interfaces; hexagonal, h110i growth axis and {001} plus {111} interfaces; hexagonal, h11 " 2 2i growth axis and {111} plus f1 " 3 31g interfaces; hexagonal, h111i growth axis and {110} interfaces; hexagonal, h111i growth axis and f11 " 2 2g interfaces. Our extensions provide the exact crystallographic description of zb-NWires with arbitrary cross sections as encountered in experiment, and thus are only limited in their precision by measurement tolerances of the imaging technique used. As previously, the results we obtain by our analytics are the number of NWire atoms N Wire , the number of bonds between such atoms N bnd , the number of NWire interface bonds N IF and cross section areas A. We demonstrated that our analytic description is applicable with the same precision to core-shell NWires with arbitrary shape and interface orientation of the core and shell, under the constraint that they share the same orientation of their growth axis, and have an interfaces roughness below the tolerance limit of the measured interface lengths. The above results are available per core and shell section of the NWire, and internal (core-shell) interface areas are given as well. If a radial distribution of the unit-cell parameter can be provided, such data can be included for all mentioned NWire cross sections, adding further flexibility and precision to their description. The description of core-shell NWires can easily be applied to multiple core-shell (layered) NWires if these comply with epitaxial growth and smooth interfaces.
The analytic description of zb-and diamond-structure NWire cross sections with arbitrarily convex shape and multiple radial layers (multiple core-shell structures) can provide major advancements in experimental data interpretation and understanding of III-V, II-VI and group-IVbased NWires. The number series allow for a deconvolution of experimental data into environment-exerted, interface-related and NWire-internal phenomena. Our method offers an essential tool to predict NWire cross sections and to tune process conditions for tailoring NWires towards desired shape and interface properties, see Kö nig & Smith (2019), Kö nig & Smith (2021) and Kö nig (2016) for details.